Why INLA?

• Bayesian Hierarchical models
  • Many latent Gaussian variables
  • Few parameters
• Monte Carlo methods impractical
• Approximate posterior marginals using Laplace expansions
  • Clever algebra, end up with posterior in denominator
  • Taylor expand log-posterior about its mode
  • Results in a Gaussian approximation
  • Some numerical integration needed

Advantages and Disadvantages

• Advantages
  • Accurate approximation
  • Fast computation of many posterior marginals
• Disadvantages
  • Does not provide full joint posterior
  • Slow with >4-6 parameters

Derivations

From Blangiardo and Cameletti (2015), Section 4.7

Include derivations of \(\tilde{p}(\boldsymbol{\psi}|\mathbf{y})\), \(\tilde{p}(\theta_{i}|\boldsymbol{\psi},\mathbf{y})\), and \(\tilde{p}(\theta_{i}|\mathbf{y})\); and explanations of mode-finding and CCD.

Simple Normal Example

Blangiardo and Cameletti (2015) section 4.9

• \(\mathbf{y} = (y_{1}, \dots, y_{n})'\) independent Gaussian observations
• \(y_{i} \sim \mathsf{N}(\theta, \sigma)\)
• \(\theta \sim \mathsf{N}(\mu_{0}, \sigma_{0})\)
• \(\psi = 1/\sigma^{2}\), \(\psi \sim \mathsf{Gamma}(a, b)\)

The posterior distribution of the nuisance parameter is

\[p(\psi|\mathbf{y}) \propto \frac{p(\mathbf{y} | \theta, \psi) p(\theta) p(\psi)} {p(\theta | \psi, \mathbf{y})}\]

Simple Normal Example

Simple Normal Example

Simple Normal Example

Simple Normal Example

Spatial Point Pattern: Trees in a Rainforest